We study Kauffman's model of folded ribbon knots: knots made a thin strip paper flat in the plane. The ribbonlength is length to width ratio such knot. show for any knot or link type that there exist constants $c_1, c_2>0$ bounded above by $c_1\cdot Cr(K)^2$, and also $c_2\cdot Cr(K)^{3/2}$. use different method each bound. constant $c_1$ quite small comparison $c_2$, first bound lower than sec...

Reducing the number of link crossings in a network drawn on plane such as wiring board is well-known problem, and especially calculation minimum crossings: this crossing problem. It has been shown that finding general solution to problem NP-hard. So, addressed for particular classes graphs also our approach paper. More precisely, we focus hereinafter torus topology. First, discuss an upper boun...

We present a linear programming based algorithm for computing a spanning tree T of a set P of n points in IR, such that its crossing number is O(min(t log n, n1−1/d)), where t the minimum crossing number of any spanning tree of P . This is the first guaranteed approximation algorithm for this problem. We provide a similar approximation algorithm for the more general settings of building a spann...

We introduce a new polynomial invariant of virtual knots and links and use this invariant to compute a lower bound on the virtual crossing number and the minimal surface genus. 1 The arrow polynomial We introduce the arrow polynomial, an invariant of oriented virtual knots and links that is equivalent to the simple extended bracket polnomial [6]. This invariant takes values in the ring Z[A,A, K...